Hardest Logic ProblemEver

[Dan]

Here is a problem that I posted a few years back, and I don't think anyone even attempted it. I made it a bit easier this time by cutting the range down from 100 to 20. Here is the problem:

There are two perfect child logicians, Sue and Pascal, that always tell the truth and can instantly deduce a valid conclusion from a set of axioms. Sue is told the sum of two numbers, x and y. Pascal is told the product of the two numbers, x and y. They also both know that 2<x<y and (x+y)<40. They have the following conversation:

Sue: Neither of us know what x or y is
Pascal: I know what x and y are
Sue: So do I

What are the two numbers?

 

[Dan]

hmm changing the range does make a difference. Take 2<x<y, (x+y)<40 as the range

 

[bbson john]

8 and 14.
It is my date of birth.




I am just randomly guessing, no idea though.

 

[Teta_Bonita]

So Sue knows x + y, Pascal knows x*y, they also both know that 1<x<y and (x+y)<40, and neither of them know what x or y is, therefore Pascal knows what x and y are?

I don't see the logic here.

Does Pascal just randomly decide that he knows what x and y are, or does that follow from the above axioms?

 

[cheddarrr]

the two numbers are:
1 and 40. duh

 

[Ennui]

41 is not less than 40.

 

[<RJMC>]

put a gun to theyr faces and force them to say the truth

 

[Dan]

So Sue knows x + y, Pascal knows x*y, they also both know that 1<x<y and (x+y)<40, and neither of them know what x or y is, therefore Pascal knows what x and y are?

I don't see the logic here.

Does Pascal just randomly decide that he knows what x and y are, or does that follow from the above axioms?

and the conversation that he has with Sue

 

[hydrometeor]

X is 4, y is 5? This seems to fit.

edit:

Nevermind, it doesn't.

 

[Raziaar]

here are two perfect child logicians, Sue and Pascal

Well duh, the answer is that nobody is perfect! It's logically impossible!

:stare:

 

[Tyguy]

Sue and Pascal are the end result of a incest filled long weekend.

 

[dfc05]

If Pascal knows the two numbers from their product, maybe it is the product of two prime numbers. Can't be the same 2 prime numbers (as in 3x3) because x<y. So try 3 and 5. But the sum of the numbers would be 8 and then Sue would have known at first that it was 3 and 5 since it can't be 2 + 6 (2<x) and it can't be 4 + 4 (x<y). So she wouldn't have said her first line.

So maybe it is the next set of prime numbers up, 3 and 7.
Then the product is 21.
The sum is 10. Sue knows then that it can't be 5+5 (x<y), and it can't be 4+6 because then that would give a product of 24, which could be either 3*8 or 4*6 (so Pascal wouldn't have known the answer). Therefore it has to be 3 and 7.

It can't be something like 5 and 11, because then Sue's sum would be 16, which could either be 3 + 13 or 5+11, and then she wouldn't have said "So do I" at the end.

I dunno why 5 and 7 wouldn't work though. Or 3 and 11. I can't bother thinking about the rest of the combinations of prime numbers.

So..... yeah. Maybe I am completely wrong.

 

[hydrometeor]

Are any of these the answer?

11 13
11 17
11 19
11 21
11 25
15 17
15 21
17 19

 

[dfc05]

Does it matter that Sue says "x OR y" in the first line instead of "x AND y"???

Does that mean that at first Pascal doesn't know what x and y are, and then after she says that, then he knows??

what the heck is going on here :O

 

[ericms]

So it is two prime numbers whose sum has at least two different possibilities (in line with all the axioms) of which Sue can only figure out are prime after consulting Pascal.

Wouldn't that contradict Pascal not knowing the two numbers however (which Sue says)? Since he would always know the two numbers.

 

[Dan]

Okay, a lot of you are having problems getting the premise of the problem so I will spell it out for you in plain language. When sue says: Neither of us know x or y, (x and y is the same because if you know one and you know the sum or the product you could figure out the other) what that means is that the sum of the numbers is such that it is impossible for any of the possible products to be the product of only that pair. Take the number 23 for instance as the sum (I am not saying that it is 23). The possible pairs of numbers which can add to 23 (within the range given) are 3+20, 4+19, 5+18, 6+17, 7+16, 8+15, 9+14, 10+13, 11+12. You will notice that none of these pairs are both prime numbers, therefore, if Sue had 23 as the sum, she would be able to say that Pascal does not know the product with absolute certainty. Comprende?

 

[ericms]

Just one thought (I'm not in the mood to calculate these things). The numbers would have to be the product of 3 or more primes (to make a prime and a composite or two composites) to resolve the issue of Pascal already knowing the numbers. Like in the case Dan just made the sum can never be that of 2 prime numbers since otherwise Pascal would know them.

 

[tehsolace]

George Washington?

 

[Acepilotf14]

This isn't logic, this is math!
And math sucks. Always.

 

[Dan]

math is logic

 

[Cole]

Haha I get it.
Neither of them were told what x or y were.
However both of them were told what x and y are.
Essentially it's an english logic question. They were both told "x and y". Neither were told "x or y" were.

haha i'm probably wrong.

I'd also like to point out. In there conversation they never told eachother the sum or the product. Therefor we can conclude that they never told eachother what they new. They were each told bits and pieces of information, then had a 3 line convo.

 

[soulslicer]

i saw it, and my brain exploded

 

[Dan]

Haha I get it.
Neither of them were told what x or y were.
However both of them were told what x and y are.
Essentially it's an english logic question. They were both told "x and y". Neither were told "x or y" were.

haha i'm probably wrong.

I'd also like to point out. In there conversation they never told eachother the sum or the product. Therefor we can conclude that they never told eachother what they new. They were each told bits and pieces of information, then had a 3 line convo.

yeah, you are wrong. They were neither told what x and y are or what x or y are. It says right in the problem, Pascal is told the product of x and y, Sue is told the sum of x and y. Although they don't tell each other the sum or the product directly, they indirectly reveal it by what they say.

 

[bam23]

I would rather watch 2 girls 1 cup 40 times.

DISCLAIMER: I AM LYING.

 

[Cole]

There is an infinite amount of possible answers.

What I don't seem to get is this:
-They both go through the list of axioms and instantly draw a conclusion.

Neither of them know x and y, in other words both of there conclusions to the problem where that they did not know what x and y were.
Pascal suddenly knows, with the only new information being that Sue doesn't know.
Sue suddenly knows with the only new information being that Pascal knows the answer.

That simply makes no sense to me. How can Pascal know the answer, simply because Sue does not know the answer. How can Sue know the answer simply because Pascal knows the answer?

They also can't tell a lie. I would imagine that the first sentence is true if you take it as: neither of them know for sure what specifically x and y the answer is suppose to be. However, they both can also conclude that there is an infinite amount of valid answers which shows in the next two sentences. Once again this is probably wrong.

 

[Azner]

put a gun to theyr faces and force them to say the truth

HAHAHAHAHAH

 

[venturon]

3 and 13.

 

[Dan]

Neither of them know x and y, in other words both of there conclusions to the problem where that they did not know what x and y were.
Pascal suddenly knows, with the only new information being that Sue doesn't know.
Sue suddenly knows with the only new information being that Pascal knows the answer.

RIGHT

They also can't tell a lie. I would imagine that the first sentence is true if you take it as: neither of them know for sure what specifically x and y the answer is suppose to be. However, they both can also conclude that there is an infinite amount of valid answers which shows in the next two sentences. Once again this is probably wrong.

WRONG

There is one specific x and one specific y which is logical

 

[Dan]

Here, I will give you guys the answer, it is 4 and 13. The product is 52, the sum is 17. Because none of the sets of numbers which add up to the sum of 17 (2+15,3+14,4+13,5+12,6+11,7+10,8+9) are sets of prime numbers, Sue can accurately say that neither of them knows the two numbers. She only knows that they will be one of the sets of numbers that add to 17, and if you multiply any of the possible sets together, you get a product that will have more than 1 possible x and y.

But then once Pascal knows that the sum is such that no possible set of numbers adding to it can be a set of prime numbers, he can deduce the true sum. The product is 52, that can be either 26*2 or 13*4. 26 and 2 add to 28. 28 has as its possible pairs (2+26, 3+25, 4+24, 5+23, 6+22, 7+21, 8+20, 9+19, 10+18, 11+17, 12+16, 13+15). 5,23 and 11,17 are pairs of prime numbers. That means that IF 28 were the sum, it would be possible for the product to be 5*23 or 11*17, and that would be completely determinant, so IF the sum were 28, Sue would not be able to say with confidence that Pascal doesn't know the two numbers. The other possible sum is 13+4=17. We have already shown that 17 doesn't have any possible pairs of prime numbers. That is how Pascal knows that 17 is the sum, and thus 4 and 13 are x and y.

Knowing that Pascal knows, Sue now knows that the product was such that he was able to determine x and y only from what she said before. That means that she knows that the product is such that one and only one set of factors adds to a sum which is not composed of any pairs of prime numbers. She can check this out by going through the possible products from her sum (2*15,3*14,4*13,5*12,6*11,7*10,8*9) and checking the other factors of them. Those are:
(The number in brackets is the sum of that set)
2*15(17) = 6*5(11) = 10*3(13)
3*14(17) = 6*7(13) = 2*21(23)
4*13(17) = 2*26(28)
5*12(17) = 10*6(16) = 15*4(19) = 20*3(23)
6*11(17) = 3*22(25)
7*10(17) = 14*5(19) = 35*2(27)
8*9(17) = 4*18(22) = 2*36(38) = 3*24(27)

The sums which have a prime number as one of their pairs are shown in bold. Because of Sue's first statement, she knows that Pete deduced that the sum cannot be one of those bolded numbers. 4 and 13 is the only set of numbers which only has bolded sums beside it. That means that 4 and 13 is the only set of numbers that could be deduced from the fact that the sum does not have any set of prime numbers. So now Sue knows what x and y are too.

 

[tehsolace]

I still think the correct answer is George Washington...